My Viewers Are Math Geniuses

This is episode "1B" in a series called "Brainstorm."

The topics for these videos are puzzle-style problems, broken into two parts each, a couple months apart.
-- In the first part, I introduce the problem, and you (the viewers) try to solve the problem yourself, or at least come up with some initial approaches or first steps.
-- In the second part, I follow up with animated discussions of the solutions that were sent to me, mixed in with my own solution.

This one is about a weird random walk process, and the question is very simply stated yet I found it quite tricky to prove the answer.

Thanks so much to everyone who submitted something! (and to everyone, in general, for watching the video)

Part 1:    • I dare you to try to solve this math ...  

These animations are made using Manim, by 3Blue1Brown.